US9972176B2 - Methods and systems for planning evacuation paths - Google Patents
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- US9972176B2 US9972176B2 US15/214,105 US201615214105A US9972176B2 US 9972176 B2 US9972176 B2 US 9972176B2 US 201615214105 A US201615214105 A US 201615214105A US 9972176 B2 US9972176 B2 US 9972176B2
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- G—PHYSICS
- G08—SIGNALLING
- G08B—SIGNALLING OR CALLING SYSTEMS; ORDER TELEGRAPHS; ALARM SYSTEMS
- G08B7/00—Signalling systems according to more than one of groups G08B3/00 - G08B6/00; Personal calling systems according to more than one of groups G08B3/00 - G08B6/00
- G08B7/06—Signalling systems according to more than one of groups G08B3/00 - G08B6/00; Personal calling systems according to more than one of groups G08B3/00 - G08B6/00 using electric transmission, e.g. involving audible and visible signalling through the use of sound and light sources
- G08B7/066—Signalling systems according to more than one of groups G08B3/00 - G08B6/00; Personal calling systems according to more than one of groups G08B3/00 - G08B6/00 using electric transmission, e.g. involving audible and visible signalling through the use of sound and light sources guiding along a path, e.g. evacuation path lighting strip
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
- G06Q10/00—Administration; Management
- G06Q10/04—Forecasting or optimisation specially adapted for administrative or management purposes, e.g. linear programming or "cutting stock problem"
- G06Q10/047—Optimisation of routes or paths, e.g. travelling salesman problem
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
- G06Q90/00—Systems or methods specially adapted for administrative, commercial, financial, managerial or supervisory purposes, not involving significant data processing
- G06Q90/20—Destination assistance within a business structure or complex
- G06Q90/205—Building evacuation
Definitions
- the embodiments herein generally relate to planning evacuation paths, and more particularly to systems and methods involving probabilistic behavior models.
- Evacuation planning is a critical aspect that involves movement of people away from threat or actual occurrence of hazard such as natural disasters, terrorist attacks, fires and bombs. Safe evacuation of a large number of people in a timely manner is a major challenge for building administrators.
- Linear Programming (LP) based polynomial time techniques for evacuation problem uses time-expanded graphs for the network, where the expression for time complexity makes it non-scalable even for mid-sized networks.
- Capacity Constrained Route Planner (CCRP) techniques use generalized shortest path technique to find shortest paths from any source to any sink, provided that there is enough capacity available on all nodes and edges of the path. Space complexity and unnecessary expansion of source nodes in each iteration are two main disadvantages of CCRP.
- CCRP++ runs faster than CCRP but the quality of solution is not good, because availability along a path may change between the times when paths are reserved and when they are actually used.
- Network flow based approaches are based on minimum cost transshipment and earliest arrival transshipment.
- the minimum cost approach does not consider the distances between evacuees and exits. It may fail if there are exits very far away. Problems also arise if a lot of exits share the same bottleneck edges.
- the earliest arrival approach uses an optimal flow over time and thus does not suffer from these problems. But the exit assignment computed by the earliest arrival approach may not be optimal.
- Embodiments of the present disclosure present technological improvements as solutions to one or more of the above-mentioned technical problems recognized by the inventors in conventional systems.
- Systems of the present disclosure identify shortest paths from at least one source to a sink in increasing order of transit time in each iteration till the transit time of the currently identified path exceeds the combined evacuation time CET of the previously added set of paths. Also, systems of the present disclosure are not required to identify all possible paths from a source to a sink; if a path is added in any iteration, it remains till the end. The combined evacuation time CET after each iteration will be monotonically non-increasing.
- system of the present disclosure includes a simple randomized behavior model to obtain a minimum expected evacuation time.
- Methods and systems are described that enable planning of evacuation paths, in a region of interest, from source nodes to sink nodes in a network of routes including a plurality of nodes (n), vertices and edges (E) therebetween.
- a computer implemented method of the present disclosure includes receiving input parameters comprising layouts of the region of interest, number of evacuees (p), transit time (T) and maximum capacity (C) associated with each edge (E); defining the network of routes based on the received input parameters; iteratively performing the steps of identifying shortest paths (P) from each source to a sink in increasing order of transit time associated with the routes constituting the network of routes, eliminating at least one of a node or an edge associated with each identified path to obtain a residual network of routes, and computing combined evacuation time (CET) and reserve capacity at each of the nodes and edges in the residual network of routes, until at least one termination condition is satisfied; and computing number of evacuees to be distributed along a each of the identified paths.
- input parameters comprising layouts of the region of interest, number of evacuees (p), transit time (T) and maximum capacity (C) associated with each edge (E); defining the network of routes based on the received input parameters; iteratively performing the steps
- the method described herein above further includes re-distributing evacuees according to a probabilistic behavioral model.
- the step of computing number of evacuees satisfies the relationship
- T i + x i C i - 1 C ⁇ ⁇ E ⁇ ⁇ T , wherein x i is the number of evacuees along path P i having T i transit time and maximum capacity C i .
- the step of re-distributing evacuees further includes a step of computing expected evacuation time according to the relationship:
- E ⁇ [ T ] max ⁇ ( T 1 + ( 1 - ⁇ ) ⁇ n C 1 - 1 , max 2 ⁇ i ⁇ k ⁇ ( T i + ⁇ x i C i - 1 ) )
- E[T] is expected evacuation time
- T i transit time along Path P i
- (1 ⁇ ) is the probability that an evacuee follows the shortest path to the nearest sink
- ⁇ is the probability that an evacuee follows suggested path
- x i is the number of evacuees along path P i
- k is the number of identified paths.
- a system for planning evacuation paths, in a region of interest, from source nodes to sink nodes in a network of routes comprising a plurality of nodes (n), vertices and edges (E) therebetween comprising one or more processors; a communication interface device; one or more internal data storage devices operatively coupled to the one or more processors for storing: an input module configured to receive input parameters comprising layouts of the region of interest, number of evacuees (p), transit time (T) and maximum capacity (C) associated with each edge (E); a network module configured to define the network of routes based on the received input parameters; a path identifier module configured to identify shortest paths (P) from each source to a sink in increasing order of transit time associated with the routes; eliminate at least one of a node or an edge associated with each identified path to obtain a residual network of routes and further configured to compute combined evacuation time (CET) and reserve capacity at each of the nodes and edges in the residual network of routes; and an input module configured to receive input parameters comprising
- system as described herein above can further comprise a path optimizer module configured to re-distribute evacuees according to a probabilistic behavioral model.
- a computer program product for processing data comprising a non-transitory computer readable medium having program instructions embodied therein for: receiving input parameters comprising layouts of the region of interest, number of evacuees (p), transit time (T) and maximum capacity (C) associated with each edge (E); defining the network of routes based on the received input parameters; iteratively performing the steps of discovering shortest paths (P) from each source to a sink in increasing order of transit time associated with the routes, eliminating at least one of a node or an edge associated with each discovered path to obtain a residual network of routes, and computing combined evacuation time (CET) and reserve capacity at each of the nodes and edges in the residual network of routes, until at least one termination condition is satisfied; computing number of evacuees to be suggested to follow a particular discovered path; and re-distributing evacuees according to a probabilistic behavioral model.
- FIG. 1 illustrates an exemplary building graph defined on the basis of input parameters received by a system of the present disclosure
- FIG. 2 illustrates a portion of an exemplary graph showing parallel flows sent on non edge-disjoint paths
- FIG. 3 illustrates an exemplary block diagram of a system for planning evacuation paths in accordance with an embodiment of the present disclosure
- FIG. 4A through FIG. 4B is an exemplary flow diagram illustrating a computer implemented method for planning evacuation paths using the system of FIG. 3 in accordance with an embodiment of the present disclosure
- FIG. 5 is a graphical illustration of evacuation time versus number of nodes for Capacity Constrained Route Planner (CCRP) and Single source Single sink Evacuation Problem (SSEP) of the present disclosure
- FIG. 6 is a graphical illustration of run time of the technique of the present disclosure versus number of nodes for Capacity Constrained Route Planner (CCRP) and Single source Single sink Evacuation Problem (SSEP) of the present disclosure.
- CCRP Capacity Constrained Route Planner
- SSEP Single source Single sink Evacuation Problem
- SSEP Single source Single sink Evacuation Problem
- FIGS. 1 through 6 where similar reference characters denote corresponding features consistently throughout the figures, there are shown preferred embodiments and these embodiments are described in the context of the following exemplary system and method.
- FIG. 1 illustrates exemplary building graph 100 defined on the basis of input parameters received by a system of the present disclosure.
- Each edge has a capacity, which is the maximum number of people that can traverse the edge per unit time and a travel time, which is the time needed to travel from one node to another node along that edge.
- region of interest as referred to in the present disclosure pertains to region or premise of threat or actual occurrence of hazards such as natural disasters, terrorist attacks, fires and bombs.
- position detection device refers to at least one of sensors, wearable devices, mobile devices including smartphones, hand-held devices, portable devices and PDAs that can enable detection of location information pertaining to an evacuee.
- Sensors referred to in this context may be heat sensors, motion sensors or location sensors based on Wi-Fi or any other means known in the art. Accordingly, in the context of the present disclosure, the expressions “position detection device” and “sensors” may be used interchangeably to refer to device that provides location information either directly or indirectly.
- transit time as referred to in the present disclosure pertains to the sum of the transit times of all the edges in path P from source s to sink t, and is denoted as T(P).
- destination arrival time of a path is the time required by a person to move from source s to sink t using path P subject to prior reservations, and is denoted as DA(P).
- DA(P) is the sum of T(P) and any intermediate delay and DA(P) ⁇ T(P).
- Capacity of a path pertains to the minimum of the capacities of all nodes and edges present in path P, and is denoted by C(P).
- saturated node or edge as referred to in the present disclosure pertains to a node or edge on a path P when capacity of that node or edge is the same as the capacity of path P.
- Exemplary building graph 100 consists of 10 vertices and 15 edges.
- For each vertex v its name and the capacity are specified by a pair of the form (v, c(v)).
- a vertex representing an exit is represented as a square, while the others are represented as circles.
- For each edge e the capacity and the travel time are specified on the edge by the pair (c(e), d(e)).
- the goal of the system of the present disclosure is to find the exit and an optimal path (route) for each evacuee, subject to the constraint that the number of source-sink paths passing through an edge does not exceed the capacity of the edge at any unit time interval.
- the objective function that the system of the present disclosure minimizes is the total time of evacuation. This is the time at which the last evacuee is evacuated, hereinafter referred to as the evacuation time.
- system of the present disclosure can minimize time T in which a feasible flow value at least f can be sent from sources to sinks.
- FIG. 2 illustrates a portion 200 of an exemplary graph showing parallel flows sent on non edge-disjoint paths.
- ordered pair (C, T) denotes capacity and transit time of an edge.
- C i and T i denote the capacity and transit time of path P i respectively
- p denotes the number of evacuees.
- Time required to evacuate p people via a path P having transit time T and capacity C is
- the formula for CET provides an expression for evacuation time and the number of people that will be evacuated on each path. It will be understood by those skilled in the art that paths having lesser arrival time will evacuate more groups.
- This technique is known as QPER (Quickest Path Evacuation Routing).
- the technique finds all edge-disjoint paths between a single source and a single sink and orders them according to the quickest evacuation time (calculated using CET) and adds them one by one.
- the technique does not use time-expanded graphs and there is no need to store availability information at each time stamp, as only edge-disjoint paths are considered. But the technique is limited to Single source Single sink Evacuation Problems (SSEP).
- SSEP Single source Single sink Evacuation Problems
- the addition of paths is not consistent, i.e., a path added at some point of time may be removed by the technique at a latter point of time, in case removal makes the solution better.
- path P 1 along with its capacity C 1 having minimum transit time is identified and capacities of each node and path of P 1 is decreased by its capacity C 1 permanently to obtain a residual graph.
- path P 2 having minimum transit time in the residual graph is identified and so on.
- evacuation time CET can be calculated.
- a formal technique for the method of the present disclosure can be as described in Technique 1 herein below.
- Input A graph G (V, E) representing the network with designated source s ⁇ V and sink t ⁇ V. Every node v ⁇ V has an occupancy and maximum capacity. Every edge e ⁇ E has a maximum capacity and transit time. Initially, all persons are in s.
- Running time analysis of SSEP in accordance with the present disclosure In the event that paths P 1 , P 2 . . . P K are identified after execution of Technique 1, as in each step at least one edge or one node is deleted, at most m+n paths will be identified by system of the present disclosure. In each path at least one person will be evacuated. In the worst case, system of the present disclosure can identify p paths. Hence, k ⁇ min(m+n, p). As each path identification can be done in O(m+n log n) time, Technique 1 requires O(min(m+n,p)(m+n log n)) time.
- Technique 1 also finds a path after saturated nodes and edges of all previously identified path are deleted if it satisfies the conditions given in Technique 1 (line numbers 4 and 6). Each path identified by CCRP can be represented as an ordered pair of path and its group size. Technique 1 also returns a path with maximum number of evacuees who can travel by that path at a time.
- each path is identified only once in Technique 1, we can also represent each path along with the capacity as an ordered pair.
- Technique 1 described herein above can be extended to the case where there is a single source and multiple sinks.
- a super sink can be created which is connected to all the sinks of the original graph.
- the capacity and transit time of all the edges (that connect the super sink to all original sinks) are ⁇ and 0 respectively.
- E ⁇ [ T ] max ⁇ ( T 1 + ( 1 - ⁇ ) ⁇ n C 1 - 1 , max 2 ⁇ i ⁇ k ⁇ ( T i + ⁇ x i C i - 1 ) ) E[T] will be minimum when it satisfies the following equation,
- Technique 1 as described herein above can be succeeded by a formal technique for re-distributing evacuees as described in Technique 2 herein below.
- Technique 2 has an expected evacuation time of CET( ⁇ P 1 , P 2 , P 3 , . . . , P k ⁇ ) when it quits from step-2.
- Technique 2 has an expected evacuation time of
- FIG. 3 illustrates an exemplary block diagram of system 300 for planning evacuation paths in accordance with an embodiment of the present disclosure
- FIG. 4A through FIG. 4B illustrates an exemplary flow diagram illustrating a computer implemented method 400 for planning evacuation paths using the system of FIG. 3 in accordance with an embodiment of the present disclosure.
- the steps of method 400 of the present disclosure will now be explained with reference to the components of system 300 as depicted in FIG. 3 for planning evacuation paths, in a region of interest, from source nodes to sink nodes in a network of routes including a plurality of nodes (n), vertices and edges (E) therebetween.
- system 300 includes one or more processors (not shown), communication interface or input/output (I/O) interface (not shown), and memory or one or more internal data storage devices (not shown) operatively coupled to the one or more processors.
- the one or more processors can be implemented as one or more microprocessors, microcomputers, microcontrollers, digital signal processors, central processing units, state machines, logic circuitries, and/or any devices that manipulate signals based on operational instructions.
- the processor(s) is configured to fetch and execute computer-readable instructions stored in the memory.
- system 300 can be implemented on a server or in a variety of computing systems, such as a laptop computer, a desktop computer, a notebook, a workstation, a mainframe computer, a server, a network server, cloud, hand-held device and the like.
- the I/O interface can include a variety of software and hardware interfaces, for example, a web interface, a graphical user interface, and the like and can facilitate multiple communications within a wide variety of networks and protocol types, including wired networks, for example, LAN, cable, etc., and wireless networks, such as WLAN, cellular, or satellite.
- the I/O interface can include one or more ports for connecting a number of devices to one another or to another server.
- the memory may include any computer-readable medium known in the art including, for example, volatile memory, such as static random access memory (SRAM) and dynamic random access memory (DRAM), and/or non-volatile memory, such as read only memory (ROM), erasable programmable ROM, flash memories, hard disks, optical disks, and magnetic tapes.
- volatile memory such as static random access memory (SRAM) and dynamic random access memory (DRAM)
- non-volatile memory such as read only memory (ROM), erasable programmable ROM, flash memories, hard disks, optical disks, and magnetic tapes.
- ROM read only memory
- erasable programmable ROM erasable programmable ROM
- flash memories hard disks, optical disks, and magnetic tapes.
- hard disks hard disks
- optical disks optical disks
- magnetic tapes magnetic tapes
- input parameters including layouts of the region of interest, number of evacuees (p), transit time (T) and maximum capacity (C) associated with each edge (E) are received at input module 10 of system 100 for planning evacuation paths.
- details of the region of interest 22 can include layouts of the region of interest as an input to system 100 .
- position detection device(s) 24 can provide location Information pertaining to each of the evacuees either directly or indirectly to input module 10 .
- system 300 of the present disclosure can be triggered upon receiving an input from one or more hazard detection device(s) 26 to initiate the planning of evacuation paths and computing number of evacuees to be suggested to follow a particular path based on method 400 of the present disclosure.
- network module 12 can define a network of routes based on the received input parameters.
- the received input parameters are defined in the form of a graph as illustrated in exemplary building graph 100 of FIG. 1 .
- path identifier module 14 can be configured to identify shortest paths (P) from each source to a sink in increasing order of transit time associated with the routes at step 406 . Further, at step 408 , path identifier module 14 can eliminate at least one of a node or an edge associated with each identified shortest path to obtain a residual network of routes. Furthermore, at step 410 , path identifier module 14 can compute combined evacuation time (CET) and reserve capacity at each of the nodes and edges in the residual network of routes. In an embodiment, path identifier module 14 is further configured to compute the CET as a function of maximum capacity (C) the transit time (T) and the number of evacuees (p).
- C maximum capacity
- T transit time
- p the number of evacuees
- termination module 20 checks for at least one termination condition to be satisfied for terminating iterative steps 406 through 410 .
- the termination conditions include checking whether there is a sink node reachable from a source node, the transit time of identified path is less than combined evacuation time (CET) and whether there is a path identified for each evacuee.
- CET combined evacuation time
- evacuee distributing module 16 can be configured to compute number of evacuees to be suggested to follow a particular identified path at step 418 when termination module 20 does not detect any of the termination conditions. In an embodiment, evacuee distributing module 16 is further configured to compute the number of evacuees such that the relationship
- T i + x i C i - 1 CET , is satisfied, wherein x i is the number of evacuees along path P i having T i transit time and maximum capacity C i .
- path optimizer module 18 can be configured to re-distribute evacuees according to a probabilistic behavioral model at step 420 .
- path optimizer module 18 can be further configured to compute the expected evacuation time according to the relationship:
- E ⁇ [ T ] max ⁇ ( T 1 + ( 1 - ⁇ ) ⁇ n C 1 - 1 , max 2 ⁇ i ⁇ k ⁇ ( T i + ⁇ ⁇ ⁇ x i C i - 1 ) )
- E[T] is expected evacuation time
- T i transit time along Path P i
- (1 ⁇ ) is the probability that an evacuee follows the shortest path to the nearest sink
- ⁇ is the probability that an evacuee follows suggested path
- x i is the number of evacuees along path P i
- k is the number of identified paths.
- the step of re-distributing evacuees is preceded by the step of receiving location information pertaining to each of the evacuees.
- system 300 can further include annunciator module 30 that can be configured to provide at least one of audio and video annunciations pertaining to suggested identified path for each evacuee.
- the SSEP and CCRP techniques were executed on a Dell Precision T7600 server having an Intel Xeon E5-2687 W CPU running at 3.1 GHz with 8 cores (16 logical processors) and 128 GB RAM.
- the operating system was Microsoft Windows 7 Professional 64-bit edition.
- FIG. 5 is a graphical illustration of evacuation time versus number of nodes for CCRP and SSEP of the present disclosure
- FIG. 6 is a graphical illustration of run time of the technique of the present disclosure versus number of nodes for CCRP and SSEP of the present disclosure. From FIG. 5 , it can be seen that the evacuation time of SSEP of the present disclosure is at the most that of CCRP. It is evident from FIG. 6 that the running time of SSEP of the present disclosure is much lower than that of CCRP. Hence, for all these instances SSEP clearly outperforms CCRP with respect to both evacuation time and run time. The absolute and relative amount by which SSEP performs better than CCRP is shown in Table I herein above.
- the description is directed towards evacuation from a premise, the invention can be applied suitably to other applications involving network flow problem including trading, traffic routing, and the like in either a closed or open region.
- the hardware device can be any kind of device which can be programmed including e.g. any kind of computer like a server or a personal computer, or the like, or any combination thereof.
- the device may also include means which could be e.g. hardware means like e.g. an application-specific integrated circuit (ASIC), a field-programmable gate array (FPGA), or a combination of hardware and software means, e.g.
- ASIC application-specific integrated circuit
- FPGA field-programmable gate array
- the means can include both hardware means and software means.
- the method embodiments described herein could be implemented in hardware and software.
- the device may also include software means.
- the invention may be implemented on different hardware devices, e.g. using a plurality of CPUs.
- the embodiments herein can comprise hardware and software elements.
- the embodiments that are implemented in software include but are not limited to, firmware, resident software, microcode, etc.
- the functions performed by various modules comprising the system of the present disclosure and described herein may be implemented in other modules or combinations of other modules.
- a computer-usable or computer readable medium can be any apparatus that can comprise, store, communicate, propagate, or transport the program for use by or in connection with the instruction execution system, apparatus, or device.
- the various modules described herein may be implemented as either software and/or hardware modules and may be stored in any type of non-transitory computer readable medium or other storage device.
- Some non-limiting examples of non-transitory computer-readable media include CDs, DVDs, BLU-RAY, flash memory, and hard disk drives.
- a data processing system suitable for storing and/or executing program code will include at least one processor coupled directly or indirectly to memory elements through a system bus.
- the memory elements can include local memory employed during actual execution of the program code, bulk storage, and cache memories which provide temporary storage of at least some program code in order to reduce the number of times code must be retrieved from bulk storage during execution.
Abstract
Description
wherein xi is the number of evacuees along path Pi having Ti transit time and maximum capacity Ci.
wherein, E[T] is expected evacuation time, Ti is transit time along Path Pi, (1−α) is the probability that an evacuee follows the shortest path to the nearest sink, α is the probability that an evacuee follows suggested path, xi is the number of evacuees along path Pi and k is the number of identified paths.
P 1 :s−B−C−E−G−t,C(P 1)=4,T(P 1)=19.
P 2 :s−A−C−E−F−t,C(P 2)=6,T(P 2)=23.
Paths P1 and P2 are not edge-disjoint, but common edge CE has capacity of 10. Accordingly, C(P1)+C(P2)=C(CE). So, flow can be sent through paths P1 and P2 in parallel and edge CE can be considered for all practical purposes as being two edges one having capacity 4 dedicated for P1 and other one having capacity 6, dedicated for P2.
wherein, Ci and Ti denote the capacity and transit time of path Pi respectively, and p denotes the number of evacuees. Time required to evacuate p people via a path P having transit time T and capacity C is
So, Pi is said to be the quickest path in the event that,
Input: A graph G (V, E) representing the network with designated source |
s ϵV and sink t ϵV. Every node v ϵV has an occupancy and maximum |
capacity. Every edge e ϵE has a maximum capacity and transit time. |
Initially, all persons are in s. |
Output: Evacuation route plan for each person. |
begin |
Initialize R = ø and CET = ∞. |
Initialize i ← 0. |
while (t is reachable from s) and number of discovered paths ≤ p − 1 ≤ |
do |
Find the shortest path Pi+1 from s to t in G (V, E) and let Ti+1, Ci+1 be |
its transit time and capacity respectively. |
If Ti+1 ≤ CET then |
R = ∪{Pi+1}. |
CET = CET (Si+1). |
Reduce capacity of each node and each edge of Pi+1 by Ci+1 |
V = V\ {v: v is a saturated node of Pi+1). |
E = E\ {e: e is a saturated edge of Pi+1}. |
end |
else |
break. |
end |
end |
Let R = {P1, P2, . . . , Pk} |
Send xi persons via Pi, 1 ≤ i ≤ k, where |
|
end |
-
- 1. Number of iterations that will return path Pi is Tk−Ti+ε, 1≤i≤k, where ε denotes number of iterations that returns path Pk.
- 2. Number of Iterations that will return path Pi before phase j is Tj−Ti, where i≤j≤k.
- 3. The same paths will be returned by Technique 1.
where p is the number of evacuees, n is the number of nodes in the graph, τ is the maximum transit time of any edge and k is the number of paths used by CCRP (and Technique 1). Technique 1 also finds a path after saturated nodes and edges of all previously identified path are deleted if it satisfies the conditions given in Technique 1 (line numbers 4 and 6). Each path identified by CCRP can be represented as an ordered pair of path and its group size. Technique 1 also returns a path with maximum number of evacuees who can travel by that path at a time. As each path is identified only once in Technique 1, we can also represent each path along with the capacity as an ordered pair. The above upper bound is tight. For instance, for a line graph of 10 nodes where first node is the source and last node is the sink, if each node and edge has capacity of 1 and each edge has a transit time of 10 and in the event there is only one evacuee, the evacuation time is 90 for p=1, k=1, n=10, τ=10.
Expected time at which last person will arrive at destination via Pi is
If the expected evacuation time in this scenario be E[T].
E[T] will be minimum when it satisfies the following equation,
Where Σi=1 nxi=n and xi≥0, ∀i. Solving the above equation we get,
Expected evacuation time given by equation (3) doesn't depend on α. This is true and solution is feasible as long as x1≥0. But it is not always the case, specifically when (1−α) Σi=2 kxi>C1(T−T1+1). So, implicitly evacuation time is dependent on α.
-
- 1) Run Technique 1. Find CET and x1, x2, . . . , xk using Equation (2).
- 2) If x1≥0 then quit; else go to step 3 of Technique 1. In this case the expected evacuation time=CET.
- 3) Assign x1′ to 0 and
-
- In this case, the expected evacuation time is
when it quits from step-3.
and technique runs in O(min(n, p)·n log n) time.
is satisfied, wherein xi is the number of evacuees along path Pi having Ti transit time and maximum capacity Ci.
wherein, E[T] is expected evacuation time, Ti is transit time along Path Pi, (1−α) is the probability that an evacuee follows the shortest path to the nearest sink, α is the probability that an evacuee follows suggested path, xi is the number of evacuees along path Pi and k is the number of identified paths. In an embodiment, the step of re-distributing evacuees is preceded by the step of receiving location information pertaining to each of the evacuees.
Number | Number | SSEP (present | Improvement in SSEP over | |
of | of | disclosure) | CCRP | CCRP (CCRP/SSEP) |
Nodes | Evacuees | EVACUATION | RUN | EVACUATION | RUN | EVACUATION | RUN | |
(n) | (p) | TIME | TIME | TIME | | TIME | TIME | |
100 | 3000 | 68 | 0.124 | 69 | 1.326 | 1.01 | 10.69 | |
were used to implement the techniques and netgen was used to generate synthetic graphs. The number of vertices in the graph varies from 100 to 500,000. The number of people varies from 3,000 to 120,000. The results are shown in Table I.
TABLE I |
Comparison between evacuation time |
and run time between SSEP and CCRP |
500 | 5000 | 130 | 0.358 | 130 | 2.73 | 1.00 | 7.63 |
1000 | 7000 | 155 | 1.014 | 156 | 14.586 | 1.01 | 14.38 |
1500 | 9000 | 115 | 1.466 | 117 | 35.443 | 1.02 | 24.18 |
2000 | 15000 | 661 | 1.622 | 661 | 29.016 | 1.00 | 17.89 |
2500 | 25000 | 179 | 2.761 | 186 | 25.739 | 1.04 | 9.32 |
5000 | 40000 | 903 | 3.899 | 903 | 93.521 | 1.00 | 23.99 |
10000 | 65000 | 517 | 12.012 | 520 | 231.535 | 1.01 | 19.28 |
15000 | 95000 | 1848 | 14.025 | 1853 | 336.946 | 1.00 | 24.02 |
25000 | 100000 | 1126 | 23.134 | 1128 | 815.682 | 1.00 | 35.26 |
50000 | 120000 | 1436 | 46.69 | 1446 | 1684.217 | 1.01 | 36.07 |
100000 | 110000 | 1032 | 93.4952 | 1044 | 3016.3005 | 1.01 | 32.26 |
500000 | 100000 | 1698 | 344.341 | 1720 | 11363.253 | 1.01 | 33.00 |
Claims (17)
E□[T]=max□(T1+(1−α)□nC1−1,max2≤i≤k□(Ti+α□□xi Ci−1))
Ti+xiCi−1=CET,
E□[T]=max□(T1+(1−α)□nC1−1,max2≤i≤k□(Ti+α□□xiCi−1))
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